Adaptivity with moving grids
Adaptivity with moving grids
Adaptivity with moving grids
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28 C. J. Budd, W. Huang and R. D. Russell<br />
If the function P in (2.26) is taken to be<br />
P =<br />
( M<br />
ξ x<br />
) 2<br />
,<br />
then we obtain MMPDE5 (see Section 3 for an alternative derivation),<br />
∂x<br />
∂t = 1 (<br />
∂<br />
M ∂x )<br />
. (2.28)<br />
τ ∂ξ ∂ξ<br />
This equation can be used to evolve the one-dimensional mesh towards<br />
an equidistributed state. It also has a natural generalization to the PMA<br />
equation derived from the optimally transported meshes we will consider in<br />
Section 3.<br />
2.6.3. Variational methods in higher dimensions<br />
Motivated by (2.27) we can consider a generalization to two dimensions,<br />
which is essentially a form of equidistribution in each coordinate direction<br />
(Huang and Russell 1997b, 1999). This is given by<br />
I(ξ,η) = 1 ∫<br />
[<br />
∇ξ T M −1 ∇ξ + ∇η T M −1 ∇η ] dx dy, (2.29)<br />
2 Ω P<br />
where M is now a symmetric positive definite matrix-valued monitor function,<br />
which is a generalization of the original scalar monitor function. The<br />
Euler–Lagrange equations which define the coordinate transformation at<br />
the steady state are then given by<br />
∇ · (M −1 ∇ξ) =0, ∇ · (M −1 ∇η) =0, (2.30)<br />
where all derivatives are expressed in terms of the physical variables so that<br />
∇ =(∂ x ,∂ y ) T . A <strong>moving</strong> mesh PDE can then be obtained via the gradient<br />
flow equations, given by<br />
∂ξ<br />
∂t = −P δI<br />
τ δξ , ∂η<br />
∂t = −P δI<br />
τ δη , (2.31)<br />
and we will give more details of this procedure in Section 3. A special case<br />
of this system is given by<br />
M = wI, (2.32)<br />
where w is known as the (scalar) weight function. This corresponds to onedimensional<br />
equidistribution and in the steady state gives the equations<br />
( ) ( )<br />
1 1<br />
∇ ·<br />
w ∇ξ =0, ∇ ·<br />
w ∇η =0. (2.33)<br />
Finding a mesh which satisfies this is called Winslow’s variable diffusion<br />
method (Winslow 1967, 1981).